面積空間に於けるcontravariantなderivativeとその応用について
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(2) Vol. 7, No. 2 Journal of Hokkaido Gakugei University Dec., 1956. On the Contravariant Derivatives in an Areal Space and Their Applications to the Space Simpei Y A NO The Study of Mathematics, Iwamizawa Branch, I-Iokkaido Gakugei University. ^ff:f3F : SftSH^^H-S contravariant ^ derivative fc^-OfEffl^-^l^-r. § 0. Introduction. It is well known that Finsler spaces C-O, C3~]u and Cartan spaces C.2') are considered as special areal spaces with conditions of m=l and m=n— 1 respectively, and there is dnality between them C^3. A Cartan space is to be regarded as a manifold of n- 1 diinensional contact of a hyperplane element ui, and then every component of geometncal objects is expressed by a function of the fundamental function F and its derivatives by xl and tti. In an areal space, if an intrinsic function ¥ of ply can be written in a function of the plane element uh-'-im but its partial derivatives of W by utf-tm have no meaning C83> L.3D- This fact raises a difficulty in the studies of the space, and to taking away this difficulty Dr A. Kawaguchi introducted the intrinsic derivatives 'F;^,^- In the previous papers C-?4J, ^15^ the present author used the differentiation to determine tlie metric tensor gi.tmi • jma and connection parameters F*ff,.. In § 2 of the present paper we take a contravariant derivative 1F.fr--i>, for the duality of the derivative W:iw In § 3 a metric tensor gtr--i\ • jvh is determined by an application of the differentiation to the fundamental function F. It is proved that an areal space having m=n—l coincide with a Cartan space in § 4. § 1. Fundamental relations. In an areal space, the plane element p1^ is considered as a contravariant vector under the coordinate transformations and an ^-covariant vector under the parameter transformations, viz.. (-Li) Xt-X'W, u?^T,\u~),. (7. 2) /^ = 3x'/3xj • ^uft/'9Uapi where i,j,k,--- =l,2,-.',n, a,;3,;-,... =2,2,...,m, n-m==^>0.. 1) The number in braket refers to the references at end of present paper.. 2) See C10J § 2 and [113 § 1 (1.32)..
(3) On the contravariant derivatives in an areal space and their applications to the space. The partial derivatives F,? of F with respect to p\ are components of a covariant vector and an zc-contravariant vector, satisfying the relations. 0.3) pfp^Sf, PtlJ=F-lF^-pfp^, where pf=F-lF^. If we take fundamental co- and contra-variant /^-vectors in the following relations respectively (7.4) »h...,A= ('i0-'^...^+l •••?:,», ^•••^= 6z'"-^Li"^ ,. then we have (1.5) uh-i^Pi =un-^, ^••"'^Pi = -uiv-i^. These /(-vectors are transformed under the coordinate transformations (-?.-?) as follows (7.6) uh...i^=.'^-l'9xji/oxtr--3xh/3xiwji-^, ui\-i\=^-l^"Qxti/'9xh"-'9xi^/'QXhuh-h,. where J== | Qx/'Sx | , A'= \ 'Qu./'Qu \ .. Considering (2. 5) and (2. 6) we have Theorem 1. 1. ;(,;i...^ /s a covariant ^-vector density of lueight 1 and u-scalar. density of weight —1 and is homogeneous of degree 1 in fl ivhile uivh is a contravariant ^-vector density of lueight —1 and n-scalar density of zueight 1 and is homogeneous of degree -I in p^. ; 2). From the relation sii"'2'"I''"i+l"''!'»'n"-i'»i/»«+r--/,i=sm!fc;"'+I".'.;" we have. (7.7) ti.n...i^uh-i^='l.. § 2. Intrinsic contravariant derivatives. Let us consider a intrinsic function W of p'-^ which is homogeneous of degree p in p'^ and a scalar density of weight <u under the coordinate transformations, viz. (2.2) g'?p'p=p'!8$, 'f(f,x~)-^'"iP^p,x'), where p and w are rational numbers. The intrinsic contravariant derivatives y).tvi\ of V are defined by. (2. 2) ^...^ = ^...<» ^^^-•p^ -(m-l^pW.-h. Multiplying both sides of (2. 2~) Un...^ and summing for ; we have, on making use of. (7. 7) and (2. 2), (2.3) WM-iwn...i^=pV.. Let d> be homogeneous of degree a in pl then the product ¥<!> is homogeneous of 1) We take notations !?-,?= 8'P'/a^, '?',?, 5-8';'?'/8^8^, etc. 2) See W Chapter 1, (8.4)..
(4) Simpei Yano degree p+u, therefore from the definition (2, 2~) we have (yfi)) ;»r"('A = i/<Hir--i\ + W.ivi^S. This relation tells us Theorem 2. 1. The contravariant differentiation of the sum, difference, and multiplcation of homogeneous functions of ply obey the same rules as in ordinary. differentiation. Evidently ;P°| are the components of a covariant vector density of weight CD. From this fact and Theorem 1. 1 we have ip.ti"-i\=/l'a~lQxii/9XJi.-.'9x^/'9xj\¥.Jr'-h and Theorem 2. 2. y.<'r'-''A ay"e </te components of a coniravariant vector of weight w-1.. Considering relations ¥^fptn=p'P',^ - 3r^ obtainde from (2. 2), and (2. 5) after some straight calculations we have. (2. 4) y\'r-i\fpe " (p- 2)^'.-^. Above relations mean the rightness of Theorem 2. 3. lP'tr--i\ are the homogeneous functions of degree p—1 in piy. From Theorem 2. 3 we can taken successively the contravariant derivatives )P,ir"t\, Jf'h of 3>'.n'"i\. From (2. 3) after some complicate calculations we have the following relations tf-t\ ^•••^A^, ,,;i ^r--injr--]n[ qfti ci yi •••vm~SD'i •••w'."3 •^m-e- •-.-' "~['l'^^,'^^i^"ri;^j^--l}}', F-lVa DS •••j}m-l Fm:'. CI 7r. •••p'"3 "'<\+1/^A+2'' 'ljin-li 'in' .W.A+2' ' 't^n. +^n-l~)F-lpy'F,c;\^, V ^ .^.•••PT.P], ,,,---PS\ '^A+l' t\+lt 7A+2 ^'/»t' «A+2 •' In. —(jn—l^\(jn+p—l^(lfM"-i\uh---h+pUiJv]\i.ti\---i>. — — (/ra + p — p'n")p y'uir--iAuJf'J\ j. !p,ir-i^-h- f,3r-hM-^={F-lm\m-l)eh-i^h-jny^_j}^ ,„... 't\+l' t\+s. •••Pm~\F,n^l.^,..:--pn^(jn-lYV^---^uh---i\\-cycl^ij~). 'tn-l" 'in'J\+lr}\+s lj]n ' v-'" *-' " '•" ''"' ' "^""^^-^'. Putting m=l or W=CFK in (2. 6) (C, K are constant) then the right side of the resulting equation vanishes identically and we have Theorem 2. 4. I/ «% areal space is a Finsler space, then the contravariant. differentiation is commtttaiive to the order. Theorem 2. 5. To the f'unction CFK (C, /.' are constanf) the coniravariant differentiation is committative to the order. When (2. 6~) is multiplied by uiy.i^ and summed for i the right side of the resulting equation vanishes by means of (2. 3), (2. /), and considering (2. 3) we have (2. 7) l/M-i\J\-3wii.. .,.,,= >F.h-]\M-i\Un...i^=^p-l~)y'J\-h.. § 3. Application of the derivatives. If we put L=—^--F3 then L is a homogeneous function of degree 2 in p^, therefore. 6—.
(5) On the contravariant derivatives in an area! space and their applications to the spape. from definition (2. 2~) we have (3. 7) L.tvi\=2Luir"t\.. Putting 2 and L in (2. 7) instead of p and V respectively we have (3. 2) L,tviUi.-h^Fmaeir-ineh-Jn(F,^ .,.tl .^., . .p.2, . .• • •pwp:".3 ) + '}\+l'l\-HrJ\+l'~t\+l f]nl~in. +2(^m+l —ms~)Littr-'twh---h.. From (3. 2~) or Theorem 2. 5 we have (3.3) LM-"i\Jr"J\^Lji-J\.,ii-t\.. Considering Theorem 2. 2 and relations (2.4) we get Theorem 3. 1. L,iv/\,jvj\ are the components of a tensor of iveight —2 and. homogeneous of degree 0 in p^. If we put G=rnn- fe.i.2 .»f.1.2.. .e.i ,»e,i .i»s.i :n...e;i ,1^ x -'Wi---iTW ""i\'"i'\''jr"j'rjvj'3 '' "'j\-"j\.1.1 .L .1 .1 -^ .2 .2 .2 .'J ^.';'n .n ^n -n.. X ( L.i'iiyi'^ Jr-'J\ L ivi\ j\"-]\ • • .£'i'"-ix; -'i'"'A), 1 (3. 5) ig<1""^ ' •?r"^ ==G~WL.Jvi\Jr-'J\,. from Theorem 3. 1 and relation €ii...,.n'9xi^/'9X]r"Qxin/'9xj»==/jsj^...j^ and (3. 3), (2. 3) and (2. 7) we have (3.6) G=z)-2'"G, and Theorem 3. 2. g'r"1'^' ^l'"^ are components of a tensor of tueight zero and skeiusymmetric luith respect to )„ subscripts i's as ivell as to j's and symmetric ivith respect to tiuo systems of these A subscripis, and homogeneous of degree zero in p^. and satisfy the relations (3. 7) g'/r-'r'A •h-h,ki-k\u^-.k\=0-. From Theorem 3. 2 we can taken gir"t\ • jvh as the components of a metric /(-tensor,. 1 and then the unit covariant /(-vector /;i...;A is determined by /2'i...;\==27'-lG21»;(;i...?\ and we have gfr"i\ - Jr--hlii...i^ljt...j^=l.. Considering Theorem 3. 1 and relations (2. 6) and (2. 3) we have 1 C-3. 8) givi\ • jr--}\.kr"k\l.j^.,.j^= — g~llir--t\gkr"k\^ where g^G~m.. After simple calculations the differential d<F is expressed by (3.9) d y= y',?^ = ?''./'i-urf;(n...^.. y _.
(6) Simpei Yano. § 4. The relations of the areal spaces and Cartan spaces. An areal space having m=n—l can be regarded as a Cartan space, for this reason we shall find out the relations betwee them. For this purpose we take m=n—l and ^=7 in this paragraph. At first, (2. 4) is written as follows (4.7) ib-Hh-in-^---]^, ;t(=£"r^«-i^...^.. Above equation means that tii coincides with hyperplane element in of Cartan spaces. From (4. 1~) and (-?. 7) we have (4.2) u,p^=0, uipa=0, uiu,=l. Putting ui==pln and u.i=p\l, we have n co- and contra-variant vector series respectively >T-. n2. .. -,-,»-!. ,->M. ni. ^. -. n;. T^f. PhPl," ',PT\Pi°, PhP^' • • ,P'n-l,Pn,. and there are relations among them (4.4) pW=oa,, a,b=l, 2,......,n.. (4. 4) tells us that pf and /^ (<3=7, 2,'",n.~) are components of linealy independent coand contra-variant n vectprs respectively. In this fact we have (4.5) PiaPa-01,. (2. 2) is written as follows V':1 =Cn-l^inl'"'n~l /?'."p2 •••pn-^-(tz-2')pff'u'. In Cartan ??rl ?2 z ;?t-i v~" ~^r - " •. space, W is considered as a homogeneous function of degree p with respects to id.. Considering W,u =W/^.'Qu^p, and -^•••t:t-^^--p^p^--^=p^+pnp'.. (not summing for a and n) we obtain, (4.6) ¥,i=9'I'~/Qu,. From this reason we have Theorem 4. 1. The covariant derivative ¥;' in the areal spaces having m=n—l, coincides iviih the partial derivative 'SVl'dti, in Carton spaces. In consequence of (4. 6~) we have ¥';'.-' '=3'2fP'/9n,i3n.j, from which we conclude. Theorem 4. 2. In the areal spaces having m=n—l, the contravariant differentiation is commzttative to the order. Putting '9SL/'9u.,3u.j=9['-1, in consequence of Theorem 4. 1. we get '^'i=L:'.j, and. (3. 4) is rewritten as G=W'£h-i»c]i-J^tl^i^"-^i^=\yiij\=^, ^ =9-l''^I »-T. From the last equations we have Theorem 4. 3. The metric tensor of the areal spaces having m=n—l coincide. tvith that of Carton spaces..
(7) On the contravariant derivatives in an areal space and their applications to the space. References. D. 2). P. Finsler :. Uber Kurven Fliichen in allgemeinen Riiumen. Dissertation, Gottingen (1918).. E. Cartan :. Les espaces mBtriques fond(ss sur la notion d'aire, Actualit'es Scicntifiqnes, 72. 3) 4) 5). 0. Veblen :. Invariants of quadratic differential forms, Cambridge Tracts, No, 24 (1933).. E. T. Davies. The theory of surfaces in a geometry based on the notion of area, Proc,. 6) 7). (1933). Les espaces de Finsler, ided, 79 (1934).. Cambridge Phil, Soc, 42 (1947). The geometry of a multiple integral, Jour, Lodon Math, Soc, 20 0945). A. Moor :. Uber die dualitiit von Finslerschen und Cartanschen Raumen, Ada, Math, 88,. (1952).. 8) A. Kawaguchi and S. Hokari : Die Grundlegung der Geometrie der »;.-dimensionalen metrischen Riiume anf Grund des Begriffs des A-dimensionalen Fliicheninhalts, Proc. Imp,. Acad., XVI C1940). 9) A. Kawaguchi : On areal spaces I, Tensor, Now Series 1, (1950). 10) On areal spaces II, Tensor, New Series 1, (1951). 11) On areal spaces III, Tensor, New Series 1, (1951). 12) A. Kawaguchi and Y. Katsurada : On areal spaces IV, Tensor, New Series 1, (1951). 13) A. Kawaguchi and K. Tandai : On areal spaces V, Toisor, New Series 2, (1952). 14) S. Yano : On the Euclidean connection in an area! space of general type (I),. Jour, Hokkaido Gakngei Umv, Set B Vol 5, No 1, (1954). 15) On the Euclidean connection in an areal speace of general type (II),. Jour, Hokkaido Gakugei Uiiiv, Set B Vol 5, No 2, 0954). 16) C. Kao: On the duality of metric spaces baced on a vector density,. Jour, Hokkaido Gakngei Umv, Set B Vol 6, No I, (1955).. 9—.
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